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MA 3502 Regression Analysis and Experimental Design
创建日期：2024-06-22 15:23:34
所属分类：数学mathematics
标签： MA 3502
Regression Analysis and Experimental Design

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MA 3502 Regression Analysis and Experimental Design

Structure of Examination Paper:

There are Five pages.

There are FOUR questions in total.

There are no appendices.

The mark obtainable for each question is 25.

Instructions to Students :

Answer FOUR questions.

Show clearly all your workings.

In all exercises, dj is the j-th digit of your student number.

Question 1.

1a. Consider the linear regression model Y = Xθ+ε, where θ is an m-vector of unknown parameters, Y is an N-vector of observations, X is a design matrix of size N × m and ε is a vector of normal random errors with mean 0 and covariance matrix σ2IN . Here σ2 is some positive unknown constant and IN is the identity matrix of size N × N. Assume that rank(X) = p ≤ m.

(i) Give a geometrical interpretation of the LSE, the least squares estimator (LSE) of θ.

(ii) Defifine the lasso estimators of θ and discuss a motivation behind these estimators.

(iii) Defifine the ridge estimator of θ and brieflfly discuss its properties.

(iv) Let Z be a vector of size m such that the linear form ZT θ is estimable. Explain the method of constructing confifidence intervals for ZT θ.

(v) Derive the maximum likelihood estimator for σ2.

[10=2+2+2+2+2]

1b. Assume m = 3, N = 5 and the model yj = θ0 + θ1xj + θ2x2j + εj . The table below displays the design points and the corresponding results:

xj 0 u −u v −v

yj 0 1 −1  1 −1

Here u = d4 + 1 and v = d4 + 2, where d4 is the fourth digit of your student number.

(i) Test the hypothesis that the complete regression model is statistically signifificant on the 95% confifidence level. Write down the corresponding ANOVA table and compute R2.

(ii) Test the hypothesis

on the 95% confifidence level.

To answer this question please use 19.0 as an approximate value for the 0.95-quantile of the F-distribution with 2 and 2 degrees of freedom.

[15=7+8]

Question 2.

2a. Consider the linear regression model Y = Xθ+ε, where θ is an m-vector of unknown parameters, Y is an N-vector of observations, X is a design matrix of size N × m and ε is a vector of normal random errors with mean 0 and covariance matrix σ2IN .

In a particular application, m = 4, N = 5 and the model is Y = θ1X1 + θ2X2 + θ3X3 + θ4X4 + ε. The table below displays the design points and the corresponding results

X1 u   -1  1  0 -1

X2 −v   1  0  1 -1

X3 u−v  0  1  1 -2

X4 u+2v  -3  1 -2 1

Y   1  1  1 -1  0


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MATH 239 QF5203 INFR08010 IT114105 FIT2093 ECON3124 ECON1195 MATH 337 STAT 3006 COMP202 COMP3130 MATH3609 COMP 250 CS6140 IEEE ELE00041I FIN40090 CIV 5899 MATH3066 IGNMENT MATH 2009 STAT3405 HIST 101 MATH 3281 CEGE0042 ICT532 LAB 7 LAB 5 MCO455 CSIT214 Q4 IFB104 A1 ECO220Y5Y DA5020 CMPSC473 FIN9007 DSCI553 MSCI534 COMP3308 CSE-112 CMPT 310 EE104 F 2C P20 ALY2010 BUSM060 STA258 ECO00030M R BIOSTAT 274 COS6012-B FIT3155 CMPUT 340 EC421 INFS4205 ECOM135 NATS 1740 COMP809 MBE 6005 STA465 CSSE2310 CIVL2700 DATA4700 CMPSC 473 ISE 529 CS5044 MAST30020 COMP2207 ENGG1001 COMP0046 FIT3165 STAT GR5241 MECH0007 vATHK1001 ANALYTIC ATHK1001 MATH352001 HW3 S1 BMAN 24662 COMP 636 Stat-461 STAT 302 STAT 2132 STAR534 ENG4137 CMP3752M CS3103 BISM3222 STA 135 ME155A IIMT3601 ECON3371 ECON 3371 HW4 FINA 5840 DT211C CTEC3902 MAT005 MATH2148 S261F Econ 332 MATH3734 MIS41270 7CCMFM13 SYST 664 CIS434 CENG10014 MT4111 MT4527 ECMM424 CMT219 Math 136 MATH 3120 COMP10002 MAST20009 MATH0058 BEEM117 ISYE 6420 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